Why is a projective representation of hyberbolic geometry convex towards the sphere's center?
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Yes. With hyperbolic geometries you want to project hemispheres to 2D for measurement, etc. Generally in physics you learn Minkowski space ( http://en.wikipedia.org/wiki/Minkowski_space ) but there are also conformal and hemispherical projections. Conformal is a plane orthogonal to the plane bisecting the sphere into northern and southern hemispheres. Hemispheric is a stereographic projection from the southern pole.
A 3D "Euclidian" model of the sphere "squashing" the hyperbolic geometries has the hemispheric projection act as a convex "plate" directed towards the center of the sphere. My question is why.